Note that conventional extremal inequality approach based on enhancement and perturbation might not be flexible enough to include all the situations, e.g., capacity of the MIMO Gaussian broadcast with common and private message. Indeed, Geng and Nair establish the Gaussian optimality through factorization property by regarding the product version of the original channel model [1]}. Another example is on rate-distortion-equivocation function evaluation of the vector Gaussian secure source coding. The conventional approach can had resisted solution beyond the case of \(\mu =1\)  [2]}.
